Упр.394 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. $$\int_{0}^{\frac{\pi}{2}} \frac{1}{2}\cos\left(x+\frac{\pi}{4}\right)\,dx$$
2. $$\int_{0}^{\frac{\pi}{3}} \frac{1}{3}\cos\left(x-\frac{\pi}{3}\right)\,dx$$
3. $$\int_{1}^{3} 3\sin(3x-6)\,dx$$
4. $$\int_{0}^{3} 8\cos(4x-12)\,dx$$

1. $$\int\limits_0^{\pi/2}\frac12\cos\left(x+\frac{\pi}{4}\right)\,dx =\frac12\sin\left(x+\frac{\pi}{4}\right)\Bigg|_0^{\pi/2}$$
   $$=\frac12\sin\left(\frac{\pi}{2}+\frac{\pi}{4}\right)-\frac12\sin\left(0+\frac{\pi}{4}\right)$$
   $$=\frac12\sin\frac{3\pi}{4}-\frac12\sin\frac{\pi}{4} =\frac12\cdot\frac{\sqrt2}{2}-\frac12\cdot\frac{\sqrt2}{2}=0.$$
2. $$\int\limits_0^{\pi/3}\frac13\cos\left(x-\frac{\pi}{3}\right)\,dx =\frac13\sin\left(x-\frac{\pi}{3}\right)\Bigg|_0^{\pi/3}$$
   $$=\frac13\sin\left(\frac{\pi}{3}-\frac{\pi}{3}\right)-\frac13\sin\left(0-\frac{\pi}{3}\right)$$
   $$=\frac13\sin 0-\frac13\sin\left(-\frac{\pi}{3}\right) =0+\frac13\cdot\frac{\sqrt3}{2}=\frac{\sqrt3}{6}.$$
3. $$\int\limits_1^3 3\sin(3x-6)\,dx =-\,\cos(3x-6)\Bigg|_1^3$$
   $$=-\cos(3\cdot 3-6)+\cos(3\cdot 1-6)$$
   $$=-\cos 3+\cos(-3)=0.$$
4. $$\int\limits_0^3 8\cos(4x-12)\,dx =2\sin(4x-12)\Bigg|_0^3$$
   $$=2\sin(4\cdot 3-12)-2\sin(4\cdot 0-12)$$
   $$=2\sin 0-2\sin(-12)=2\sin 12.$$

Ответ

1) $$0$$; 2) $$\frac{\sqrt3}{6}$$; 3) $$0$$; 4) $$2\sin 12$$.